Abstract

Let A be a commutative nilpotent finitely-dimensional algebra over a field F of characteristic p > 0. A conjecture of Eggert says that p. dim A(p) dim A, where A(p) is the subalgebra of A generated by elements ap , a ∈ A. We show that the conjecture holds if A(p) is at most 2-generated.

Keywords

Introduction

Let F be a field of characteristic p>0 and A a commutative (associative) nilpotent finite-dimensional algebra over F . Let A(p) be the subalgebra generated by the set {ap| a ∈ A}. N. Eggert [1] conjectured that

p· dim A(p) ≤ dim A.

This conjecture gives an answer to the problem, when a finite abelian group is isomorphic to the adjoint group of some finite commutative nilpotent F-algebra. Recall that the adjoint group of A is the set A with the operation x○ y=x + y + x y for every x, y ∈ A.

Validity of this hypothesis would also have influence on an estimation of a (Prüfer) rank of a product of two (abelian) p-groups.

Another type of results presented by McLean [6,7]. He showed that this conjecture is true if the algebra A is either radical of a group algebra of a finite abelian group or A is graded and at least one of the following conditions is fulfilled:

(i) p = 2 and (A(p))4 = 0.

(ii) A(p) is 2-generated.

(iii) (A(p))3 = 0.

(iv) n < 3p and 3 ≤ s - 1 ≤ p, where n is the number of generators of A(p) and s is the index of nilpotence of A(p).

We also should mention the result of Gorlov [8]. He proved the conjecture for nilpotent algebras A with a metacyclic adjoint group.

One paper concerning Eggert's conjecture appeared in 2002 and the author L. Hammoudi [9] claimed he proved it. But, as Amberg [10] and McLean [7] have shown, his proof was incorrect.

In this short note we sketch out the main steps of the proof that Eggert's conjecture is true if the subalgebra A(p) has at most two generators. For the details, the reader is referred to Korbelar [11].

Since we will deal with nilpotency and commutativity only, we point out that the word 'algebra' will mean a commutative one and not necessary possesing a unit.

For an algebra A and a subset X ⊆ A we denote ([X], resp.) the algebra (vector space, resp.) generated by X.

An algebra A is called nilpotent if Am=0 for some m ∈ N.

Through this paper let always F be a field of characteristic p > 0 and R = F [x, y] be the ring of polynomials over the variables x, y and the field F.

We start with the remark, that the number of any minimal generating set of a finite generated nilpotent F -algebra A is equal to dim A/A2. This implies the following:

Lemma 1.1.Suppose that Eggert's conjecture holds for every nilpotent 2-generated F -algebra. Then it also holds for every nilpotent F -algebra A such that A(p) is a 2-generated F -algebra.

In the rest we deal with 2-generated nilpotent algebras.

Bases of Nilpotent Algebras

We will use the well-known concept of monomial ordering and standard bases.

For put

Denote the multiplicative monoid with the lexicographical ordering ≤ such that

Eggert's Conjecture for 2-generated Algebras

Let I ⊆ Rx + Ry be an ideal in R such that A = Rx + Ry/I is a nonzero nilpotent F -algebra.

We have A = and A(p) =

By definition of there are such that mare normal.

The main idea of the proof lies in the fact that taking a normal polynomial from I, dividing it by x and then multiplying by some suitable yk, we get again a member of I (3.3). Then, using binomial formula in a suitable way, we obtain a polynomial that will estimate the number (see 3.4 and the definition of BA(p)(a1p,a2p)

Lemma 3.1.

Definition 3.2. Denote

wA = max BAA(x + I, y + I).

For denote

the least integer such that Put

Following lemma is obtained using induction.

Lemma 3.3.Let 1≤ i ≤ n0 + 1 and 0 ≠ f ∈ I be such that m(f) xi. Then

The proof of the next proposition uses only the binomial formula. It finds the particular polynomial the we need to make an estimation of the numbers Di and thus of the dimension of A(p).

Proposition 3.4.

Now, only exploring carefully the previous cases for i and li we get the following interesting claim. It says that the inequality holds for almost every i.

Theorem 3.5. One of the following cases takes place:

And our main result is just an easy corollary of this and 1.1.

Theorem 3.6. Let A be a nilpotent F -algebra, char F=p>0, such that A(p) is 2-generated. Then p·dim A(p) dim A.